scholarly journals On the Real-Rootedness of the Veronese Construction for Rational Formal Power Series

2017 ◽  
Vol 2018 (15) ◽  
pp. 4780-4798 ◽  
Author(s):  
Katharina Jochemko
Keyword(s):  
Power Series ◽  
Formal Power Series ◽  
The Real ◽  
Formal Power

scholarly journals Purely periodic β-expansions in cubic Salem base in Fq((x−1))

2016 ◽  
Vol 100 (114) ◽  
pp. 279-285
Author(s):  
Faïza Mahjoub
Keyword(s):  
Power Series ◽  
Finite Field ◽  
Formal Power Series ◽  
Real Case ◽  
The Real ◽  
Formal Power

Let Fq be the finite field with q elements and ? Salem series in Fq((X?1)). It is proved in [15] that, in this case, all elements in Fq(X,?) have purely periodic ?-expansion. We characterize the formal power series f in Fq(X,?) with purely periodic ?-expansions by the conjugate vector ~f when ? is a cubic unit. No similar results exist in the real case.

SUR LE BÊTA-DÉVELOPPEMENT DE 1 DANS LE CORPS DES SÉRIES FORMELLES

2006 ◽  
Vol 02 (03) ◽  
pp. 365-378 ◽  
Author(s):  
M. HBAIB ◽  
M. MKAOUAR
Keyword(s):  
Power Series ◽  
Formal Power Series ◽  
Unit Disk ◽  
Real Case ◽  
Fixed Element ◽  
The Real ◽  
Formal Power

Let β be a fixed element of 𝔽q((X-1)) with polynomial part of degree ≥ 1, then any formal power series can be represented in base β, using the transformation Tβ : f ↦ {βf} of the unit disk [Formula: see text]. Any formal power series in [Formula: see text] is expanded in this way into dβ(f) = (ai(X))i≥1, where [Formula: see text]. The main aim of this paper is to characterize the formal power series β(|β| > 1), such that dβ(1) is finite, eventually periodic or automatic (such characterizations do not exist in the real case).

scholarly journals On transformations of formal power series

2003 ◽  
Vol 184 (2) ◽  
pp. 369-383 ◽  
Author(s):  
Manfred Droste ◽  
Guo-Qiang Zhang
Keyword(s):  
Power Series ◽  
Formal Power Series ◽  
Formal Power

scholarly journals On the Square Root of a Bell Matrix

2021 ◽  
Vol 76 (1) ◽  
Author(s):  
Donatella Merlini
Keyword(s):  
Power Series ◽  
Closed Form ◽  
Formal Power Series ◽  
Computational Method ◽  
Square Root ◽  
Riordan Arrays ◽  
Formal Power

AbstractIn the context of Riordan arrays, the problem of determining the square root of a Bell matrix $$R={\mathcal {R}}(f(t)/t,\ f(t))$$ R = R ( f ( t ) / t , f ( t ) ) defined by a formal power series $$f(t)=\sum _{k \ge 0}f_kt^k$$ f ( t ) = ∑ k ≥ 0 f k t k with $$f(0)=f_0=0$$ f ( 0 ) = f 0 = 0 is presented. It is proved that if $$f^\prime (0)=1$$ f ′ ( 0 ) = 1 and $$f^{\prime \prime }(0)\ne 0$$ f ″ ( 0 ) ≠ 0 then there exists another Bell matrix $$H={\mathcal {R}}(h(t)/t,\ h(t))$$ H = R ( h ( t ) / t , h ( t ) ) such that $$H*H=R;$$ H ∗ H = R ; in particular, function h(t) is univocally determined by a symbolic computational method which in many situations allows to find the function in closed form. Moreover, it is shown that function h(t) is related to the solution of Schröder’s equation. We also compute a Riordan involution related to this kind of matrices.

scholarly journals Two Interacting Coordinate Hopf Algebras of Affine Groups of Formal Series on a Category

Algebra ◽  
2013 ◽  
Vol 2013 ◽  
pp. 1-10
Author(s):  
Laurent Poinsot
Keyword(s):  
Power Series ◽  
Formal Power Series ◽  
Semidirect Product ◽  
Hopf Algebras ◽  
Formal Series ◽  
Product Structure ◽  
Locally Finite ◽  
Commutative Algebras ◽  
Affine Groups ◽  
Formal Power

A locally finite category is defined as a category in which every arrow admits only finitely many different ways to be factorized by composable arrows. The large algebra of such categories over some fields may be defined, and with it a group of invertible series (under multiplication). For certain particular locally finite categories, a substitution operation, generalizing the usual substitution of formal power series, may be defined, and with it a group of reversible series (invertible under substitution). Moreover, both groups are actually affine groups. In this contribution, we introduce their coordinate Hopf algebras which are both free as commutative algebras. The semidirect product structure obtained from the action of reversible series on invertible series by anti-automorphisms gives rise to an interaction at the level of their coordinate Hopf algebras under the form of a smash coproduct.

Cyclic codes over formal power series rings

2011 ◽  
Vol 31 (1) ◽  
pp. 331-343 ◽  
Author(s):  
Steven T. Dougherty ◽  
Liu Hongwei
Keyword(s):  
Power Series ◽  
Formal Power Series ◽  
Cyclic Codes ◽  
Power Series Rings ◽  
Formal Power
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